Constructing and programming quantum hardware for robust quantum annealing processes

ABSTRACT

Among other things, an apparatus comprises quantum units; and couplers among the quantum units. Each coupler is configured to couple a pair of quantum units according to a quantum Hamiltonian characterizing the quantum units and the couplers. The quantum Hamiltonian includes quantum annealer Hamiltonian and a quantum governor Hamiltonian. The quantum annealer Hamiltonian includes information bearing degrees of freedom. The quantum governor Hamiltonian includes non-information bearing degrees of freedom that are engineered to steer the dissipative dynamics of information bearing degrees of freedom.

CLAIM OF PRIORITY

This application is a continuation application of, and claims priorityto, U.S. patent application Ser. No. 17/115,493, titled “CONSTRUCTINGAND PROGRAMMING QUANTUM HARDWARE FOR ROBUST QUANTUM ANNEALINGPROCESSES,” filed on Dec. 8, 2020, which is a continuation applicationof, and claims priority to, U.S. patent application Ser. No. 16/841,251,now U.S. Pat. No. 10,915,832, titled “CONSTRUCTING AND PROGRAMMINGQUANTUM HARDWARE FOR ROBUST QUANTUM ANNEALING PROCESSES,” filed on Apr.6, 2020, which application is a divisional application of, and claimspriority to, U.S. patent application Ser. No. 16/450,461, now U.S. Pat.No. 10,614,372, titled “CONSTRUCTING AND PROGRAMMING QUANTUM HARDWAREFOR ROBUST QUANTUM ANNEALING PROCESSES,” filed on Jun. 24, 2019, whichapplication is a divisional application of, and claims priority to, U.S.patent application Ser. No. 15/109,614, now U.S. Pat. No. 10,346,760,titled “CONSTRUCTING AND PROGRAMMING QUANTUM HARDWARE FOR ROBUST QUANTUMANNEALING PROCESSES,” filed on Jul. 1, 2016, which application is theInternational Application under 35 U.S.C. 371 of WIPO Application No.PCT/US2014/072959, filed on Dec. 31, 2014, which application claimspriority under 35 USC § 119(e) to U.S. Patent Application Ser. No.61/985,348, filed on Apr. 28, 2014, and U.S. Patent Application Ser. No.61/924,207, filed on Jan. 6, 2014. The disclosure of each of theforegoing applications is incorporated herein by reference.

BACKGROUND

This specification relates to constructing and programming quantumhardware for quantum annealing processes that can perform reliableinformation processing at non-zero temperatures.

SUMMARY

Artificial intelligent tasks can be translated into machine learningoptimization problems. To perform an artificial intelligence task,quantum hardware, e.g., a quantum processor, is constructed andprogrammed to encode the solution to a corresponding machineoptimization problem into an energy spectrum of a many-body quantumHamiltonian characterizing the quantum hardware. For example, thesolution is encoded in the ground state of the Hamiltonian. The quantumhardware performs adiabatic quantum computation starting with a knownground state of a known initial Hamiltonian. Over time, as the knowninitial Hamiltonian evolves into the Hamiltonian for solving theproblem, the known ground state evolves and remains at the instantaneousground state of the evolving Hamiltonian. The energy spectrum or theground state of the Hamiltonian for solving the problem is obtained atthe end of the evolution without diagonalizing the Hamiltonian.

Sometimes the quantum adiabatic computation becomes non-adiabatic due toexcitations caused, e.g., by thermal fluctuations. Instead of remainingat the instantaneous ground state, the evolving quantum state initiallystarted at the ground state of the initial Hamiltonian can reach anexcited state of the evolving Hamiltonian. The quantum hardware isconstructed and programmed to suppress such excitations from aninstantaneous ground state to a higher energy state during an earlystage of the computation. In addition, the quantum hardware is alsoconstructed and programmed to assist relaxation from higher energystates to lower energy states or the ground state during a later stageof the computation. The robustness of finding the ground state of theHamiltonian for solving the problem is improved.

The details of one or more embodiments of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will be apparent from the description, the drawings, andthe claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic perspective view of a quantum annealing processorwithin a Chimera connectivity of interacting qubits.

FIG. 2 is a schematic diagram showing the structures and interactions oftwo qubits in a quantum processor, where the interactions include x-xand x-z interactions of a quantum governor.

FIG. 2A is a schematic diagram showing a Josephson box, including aJosephson junction and a capacitor.

FIG. 3 is a schematic diagram showing the effect of a quantum governoron transitions among instantaneous energy states during a quantumannealing process

FIG. 4 is a schematic diagram showing the interplay of an initialHamiltonian, a problem Hamiltonian, and a Hamiltonian of a quantumgovernor chosen for the problem Hamiltonian during a quantum annealingprocess.

FIG. 5 is a flow diagram of an example process for determining a quantumgovernor distribution.

FIG. 6 is a flow diagram of an example process for performing anartificial intelligence task.

DETAILED DESCRIPTION Overview

Solutions to hard combinatorial problems, e.g., NP-hard problems andmachine learning problems, can be encoded in the ground state of amany-body quantum Hamiltonian system, which is also called a quantumannealer (“QA”). A quantum annealing process at zero temperature limitis known as adiabatic quantum computation, in which the QA isinitialized to a ground state of an initial Hamiltonian H_(i) that is aknown and easy to prepare. Over time, the QA is adiabatically guidedwithin the Hilbert space to a problem Hamiltonian H_(p) that encodes theproblem. In theory, during the adiabatic quantum computation, the QA canremain in the instantaneous ground state of a Hamiltonian H_(total)evolving from H_(i) to H_(p), where H_(total) can be expressed as:

H _(total)=(1−s)H _(i) +sH _(p),

where s is a time dependent control parameter:

s=t/t _(T),

and t_(T) is the total time of the adiabatic quantum computation. The QAwill reach the ground state of the problem Hamiltonian H_(p) withcertainty, if the evolution of system is sufficiently slow with respectto the intrinsic energy scale of the system.

In reality, the quantum computation may not be completely adiabatic andthe QA may reach an excited state of H_(total) during the computation,which can lead to inaccurate result at the end of the quantumcomputation. For example, in many hard combinatorial optimizationproblems, e.g., in decision problems, when the problem Hamiltoniandemonstrates a phase transition in its computational complexity, thesize of a gap between an excited state and the ground state of H_(total)can be small, e.g., exponentially small, with respect to the intrinsicenergy scale of the system. In such situations, the QA may undergo aquantum phase transition and can reach a large number, e.g., anexponentially large number, of excited states. In addition, the QA mayalso deviate from the ground state of H_(total) due to other factorssuch as quantum fluctuations induced by environmental interactions withthe system and system imperfection errors, including control errors andfabrication imperfections. In this specification, the process of drivingthe QA from the ground state of H_(i) to the ground state of H_(p) iscalled a quantum annealing schedule or a quantum annealing process.

Quantum hardware, such as quantum processors, of this specificationincludes a quantum chip that defines a quantum governor (“QG”) inaddition to H_(i) and H_(p), such that the evolving HamiltonianH_(total) becomes H_(tot):

H _(tot) =I(t)H _(i) +G(t)H _(G) +P(t)H _(p) +H _(AG-B),

where I(t) and P(t) represent the time-dependency of the initial andproblem Hamiltonians, H_(i) and H_(p), respectively; G(t) represents thetime-dependency of the QG Hamiltonian, H_(G); and H_(AG-B) is theinteraction of the combined QA-QG system with its surroundingenvironment, commonly referred to as a bath. In a simplified example,I(t) equals (1−s), P(t) equals s, G(t) equals s(1−s), and H_(AG-B) isassumed to be non-zero but constant during the quantum annealingprocess. The strength of H_(AG-B) is related to spectral density of bathmodes that can often be characterized off-line by a combination ofexperimental and theoretical quantum estimation/tomography techniques.

Generally, the QG can be considered as a class ofnon-information-bearing degrees of freedom that can be engineered tosteer the dissipative dynamics of an information-bearing degree offreedom. In the example of H_(total), the information-bearing degree offreedom is the QA. The quantum hardware is constructed and programmed toallow the QG to navigate the quantum evolution of a disordered quantumannealing hardware at finite temperature in a robust manner and improvethe adiabatic quantum computation process. For example, the QG canfacilitate driving the QA towards a quantum phase transition, whiledecoupling the QA from excited states of H_(total) by making the excitedstates effectively inaccessible by the QA. After the quantum phasetransition, the QA enters another phase in which the QA is likely to befrozen in excited states due to quantum localization or Andersonlocalization. The QG can adjust the energy level of the QA to be in tunewith vibrational energies of the environment to facilitate the QA torelax into a lower energy state or the ground state. Such an adjustmentcan increase the ground state fidelity, i.e., the fidelity of the QAbeing in the ground state at the end of the computation, and allow theQA to avoid a pre-mature freeze in suboptimal solutions due to quantumlocalization.

Generally, the QA experiences four phases in a quantum annealing processof the specification, including initialization, excitation, relaxation,and freezing, which are explained in more detailed below. The QG canassist the QA in the first two phases by creating a mismatch betweenaverage phonon energy of the bath and an average energy level spacing ofthe

A to suppress unwanted excitations. In the third and fourth stages, theQG can enhance thermal fluctuations by creating an overlap between thespectral densities of the QA and the bath. The enhanced thermalfluctuations can allow the QA to have high relaxation rates from higherenergy states to lower energy states or the ground state of the problemHamiltonian H_(p). In particular, the QG can allow the QA to defreezefrom non-ground states caused by quantum localization.

The QG can be used to achieve universal adiabatic quantum computing whenquantum interactions are limited due to either natural or engineeredconstraints of the quantum hardware. For example, a quantum chip canhave engineering constraints such that the Hamiltonian representing theinteractions of qubits on the quantum chip is a k-local stochasticHamiltonian. The quantum hardware can be constructed and programmed tomanipulate the structural and dynamical effects of environmentalinteractions and disorders, even without any control over the degrees offreedom of the environment.

Generally, the QG is problem-dependent. The quantum hardware of thespecification can be programmed to provide different QGs for differentclasses of problem Hamiltonians. In some implementations, a QG can bedetermined for a given H_(p) using a quantum control strategy developedbased on mean-field and microscopic approaches. In addition oralternatively, the quantum control strategy can also implement randommatrix theory and machine learning techniques in determining the QG. Thecombined QA and QG can be tuned and trained to generate desiredstatistical distributions of energy spectra for H_(p), such as Poisson,Levy, or Boltzmann distributions.

Example Quantum Hardware

As shown in FIG. 1, in a quantum processor, a programmable quantum chip100 includes 4 by 4 unit cells 102 of eight qubits 104, connected byprogrammable inductive couplers as shown by lines connecting differentqubits. Each line may represent one or multiple couplers between a pairof qubits. The chip 100 can also include a larger number of unit cells102, e.g., 8 by 8 or more.

FIG. 2 shows an example pair of coupled qubits 200, 202 in the same unitcell of a chip, such as any pair of qubits in the unit cell 102 of thequantum chip 100. In this example, each qubit is a superconducting qubitand includes two parallelly connected Josephson boxes 204 a, 204 b or206 a, 206 b. Each Josephson box can include a Josephson junction and acapacitance connected in parallel. An example is shown in FIG. 2A, inwhich a Josephson box 218 includes a Josephson junction 220 parallellyconnected to a capacitance 222. The qubits 200, 202 are subject to anexternal magnetic field B applied along a z direction perpendicular tothe surface of the paper on which the figure is shown; the B field islabeled by the symbol ⊗. Three sets of inductive couplers 208, 210, 212are placed between the qubits 200, 202 such that the qubits are coupledvia the z-z, x-z, and x-x spin interactions, where the z-z interactionsrepresent the typical spin interactions of a QA, and the x-z, x-xinteractions are auxiliary interactions representing the controllabledegrees of freedom of a QG. Here x, y, and z are spin directions inHilbert space, in which each direction is orthogonal to the other twodirections.

Compared to one conventional quantum chip known in the art, the qubitsthat are coupled along the z-z spin directions in the chip 100 of FIG. 1are additionally coupled along the x-z spin directions and the x-x spindirections through the coupler sets 210, 212. The Hamiltonian of theconventional quantum chip can be written as:

$H_{SG} = {{{I(t)}{\sum\limits_{i}^{N}\;\sigma_{i}^{x}}} + {{P(t)}\left( {{- {\sum\limits_{i}^{N}\;{h_{i}\sigma_{i}^{z}}}} + {\sum\limits_{i}^{N}\;{J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}}}} \right)}}$

where σ_(i) ^(x) and σ_(i) ^(z) quantum operators that have binaryvalues and each represents the spin of the i^(th) qubit along the xdirection or the z direction, respectively. h_(i) and J_(ij) areparameters that can be programmed for different problems to be solved byadjusting the inductive coupler set 208. h_(i) and J_(ij) have realvalues. The sparsity of the parameter J_(ij) is constrained by thehardware connectivity, i.e., the connectivity of the qubits shown inFIG. 1. For unconnected qubits, the corresponding is J_(ij). Again, I(t)and P(t) represent the time-dependency of initial and problemHamiltonians, respectively. In a simplified example, I(t) equals (1−s),and P(t) equals s, where s equals t/t_(T).

The additional coupler sets 210, 212 introduce additional quantumcontrol mechanisms to the chip 100.

In general the control mechanisms of a QG acts within the same Hilbertspace of the QA and include:

(i) Site dependent magnetic field on any spin, or quantum disorders,such as σ_(i) ^(y), which is also binary and represents the spin of thei^(th) qubit along the y direction;

(ii) Two-body quantum exchange interaction terms, e.g., σ_(i) ^(x)σ_(j)^(z), that represents coupling of the i^(th) and j^(th) qubits along thex-z directions;

(iii) A global time-varying control knob G(t), which can be s(1−s),where s=t/t_(T); and

(iv) A set of macroscopic, programmable control parameters of theenvironment, such as the temperature T.

Accordingly, the Hamiltonian H_(tot) for the combined QA-QG system inthe chip 100 is:

$H_{tot} = {{{I(t)}{\sum\limits_{i}^{N}\;\sigma_{i}^{x}}} + {{G(t)}\left( {{\sum\limits_{m \in {\{{x,y,z}\}}}{\sum\limits_{i}^{N}\;{ɛ_{i,m}\sigma_{i}^{m}}}} + {\sum\limits_{m,{n \in {\{{x,y,z}\}}}}{\sum\limits_{i < j}^{N}\;{g_{ijmn}\sigma_{i}^{m}\sigma_{j}^{n}}}}} \right)} + {{P(t)}\left( {{- {\sum\limits_{i}^{N}\;{h_{i}\sigma_{i}^{z}}}} + {\sum\limits_{i}^{N}\;{J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}}}} \right)}}$

where ε_(i,m) denotes the QG induced disorders, the tensor g_(ijmn)defines the general two-body interaction parameters that specify the QG,and I(t), G(t), and P(t) are as described above. In this Hamiltonian,the initial Hamiltonian is:

${H_{i} = {\sum\limits_{i}^{N}\;\sigma_{i}^{x}}},$

the problem Hamiltonian H_(p) is:

${H_{p} = {{- {\overset{N}{\sum\limits_{i}}{h_{i}\sigma_{i}^{z}}}} + {\sum\limits_{i}^{N}\;{J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}}}}},$

and the QG Hamiltonian

is:

$H_{QG} = {{\sum\limits_{m \in {\{{x,y,z}\}}}{\sum\limits_{i}^{N}\;{ɛ_{i,m}\sigma_{i}^{m}}}} + {\sum\limits_{m,{n \in {\{{x,y,z}\}}}}{\sum\limits_{i < j}^{N}{g_{ijmn}\sigma_{i}^{m}{\sigma_{j}^{n}.}}}}}$

Again, the total Hamiltonian is:

H _(tot)=(1−t/t _(T))H _(i) +t/t _(T)(1−t/t _(T))

+(t/t _(T))H _(p).

Programming the

uantum Hardware

For a given problem and its corresponding problem Hamiltonian H_(p), aQG is determined to improve the ground state fidelity of the QA. The QGcan be determined without needing to diagonalize H_(p). Various QGrealizations can be repeated to statistically improve knowledge aboutthe computational outcomes.

In some implementations, a QG is determined such that before a systemcharacterized by H_(total) experiences a quantum phase transition, theQG Hamiltonian

acts to suppress excitations of the QA. In particular, the QG is out ofresonance with the average phonon energy of the bath, which creates amismatch between the average phonon energy and average energy levelspacing of the combined QA and QG, or H_(tot) to reduce unwantedexcitations. After the system undergoes the quantum phase transition,the QG Hamiltonian

acts to enhance relaxation of the QA from any excited state to theground state of H_(tot). In particular, the average energy level spacingof H_(tot) is in resonance with the average phonon energy. The QGenhances thermal fluctuations by creating an overlap between thespectral densities of the system and its bath. The thermal fluctuationscan facilitate the QA to reach the ground state of H_(tot) at a highrelaxation rate and prevent the QA from being prematurely frozen at anexcited state due to quantum localization.

An example of desirable QG functions is shown in FIG. 3. The energylevels E₀, E₁, E₂, . . . E_(i) (not shown) of H_(total) are plotted as afunction of time t. At t=0, H_(total) is H_(i), and at t=t_(T),H_(total) is H_(p). During a quantum annealing process from t=0 tot=t_(T), the QA approximately experiences an initialization phase fromt=0 to t=t₁, an excitation phase from t=t₁ to t=t₂, a relaxation phasefrom t=t₂ to t=t₃, and a freezing phase from t=t₃ to t=t_(T). The timet₂ can correspond to a time at which a quantum phase transition occursin a system characterized by H_(total). During the excitation phase, theQG increases, as indicated by arrows 300, 302, the average energyspacing between adjacent energy levels Δε_(i), such as Δε_(i)=E₂−E₁ andΔε₀=E₁−E₀, such that the increased energy spacing is much larger thanthe average phonon energy. During the relaxation phase, the QG adjuststhe average energy spacing Δε₀, Δε₁, . . . to be comparable to theaverage phone energy to facilitate relaxation of the QA from excitedstates to lower energy states or the ground state, as indicated byarrows 304, 306, 308, 310.

The interplay of the three Hamiltonians, H_(i), H_(p), and

over time in different phases of the quantum annealing process isschematically shown in FIG. 4. The control parameters I(t), P(t), andG(t) control the shapes of the curves for the correspondingHamiltonians. In this example, I(t) and P(t) are linear and G(t) isparabolic.

In addition, the QG can be chosen to allow the QA of H_(tot) to steadilyevolve over the QA schedule and reach a final state that has a maximumoverlap with the ground state of H_(p). Ideally, the ground statefidelity of the QA at time t_(T) is 1. However, unity fidelity is hardto achieve within a finite period of time. Other than at time 0 and attime t_(T), the QA of H_(tot) is in a mixed state of the combined H_(p),H_(i), and

. The evolution of the QA can be expressed as:

|ε₀ ^(i)

ε₀ ^(i)|→ρ_(A)(t)→|ε₀ ^(p)

ε₀ ^(p)|

where |ε₀ ^(i)

is the state of the QA at time 0, |ε₀ ^(p)

is the state of the QA at time t_(T), and ρ_(A)(t) is the densityfunction of the QA at other times. By assigning a probability, e.g.,using a probability mass function, to each state |ε₀ ^(p)

, the evolution of the QA can be further expressed as:

$\left. {\left. ɛ_{0}^{i} \right\rangle\left\langle ɛ_{0}^{i} \right.}\rightarrow\left. {\rho_{A}(t)}\rightarrow{\sum\limits_{k}{{f^{G}(k)}\left. ɛ_{k}^{p} \right\rangle\left\langle ɛ_{k}^{p} \right.}} \right. \right.$

where ƒ_(G)(k) is the probability mass function, k=0, 1, . . . , andcorresponds to quantum state levels, and Σ_(k)ƒ_(G)(k)=1. If the groundstate fidelity is 1, then ƒ^(G)(0)=1, and ƒ^(G) (k≠0)=0. As describedabove, such a unity fidelity is hard to realize. Instead, a desirable QGcan be selected to provide an exponential distribution function as:

ƒ^(G)(k,λ _(G))=

ε_(k) ^(p)|ρ_(A)(t _(T),λ_(G))|ε_(k) ^(p)

where λ_(G) defines the distribution of a QG family suitable for usewith H_(p). The probability mass function can be any probabilitydistribution function. Examples include Poisson distribution functions,Levy distribution functions, and Boltzmann distribution functions.

To determine a QG with desirable functions for a problem, includingthose functions described above with reference to FIGS. 3 and 4, one ormore techniques can be used, including, for example, open quantum systemmodels, random matrix theory, and machine learning. An example process500 for determining a QG is shown in FIG. 5, which can be performed by aclassical processor, such as a classical computer, or a quantumprocessor, or a combination of them.

In the process 500, information about energy states of a known H_(total)is obtained (502). In some implementations, a QG is constructed usingrandom matrix theory (RMT) and some predictions on general statisticalproperties of the combined QA-QG system can be made. In particular,using the random matrix theory, approximate distributions of the energylevels E_(i) of the i energy states, where i is 0, 1, 2, . . . , aspontaneous energy spectrum, the spacings Δε_(i) of the energy levels,and the average level spacing Δε of the spacings can be obtained. Insome implementations, the average energy level spacing Δε is obtainedusing mean-field theories without explicitly diagonalizing H_(total). Insome examples, path-integral Monte-Carlo is used for evaluating anapproximate ground state energy of H_(total).

In some implementations, the average energy level spacing at time t isestimated as:

$\overset{\_}{{\Delta ɛ}(t)} = \frac{2\Sigma_{i = 0}^{N - 1}{{{ɛ_{i}(t)} - {ɛ_{j}(t)}}}}{N\left( {N - 1} \right)}$

where ε_(i)(t) is the energy of the i^(th) instantaneous eigenstateenergy of H_(total), and N is the total number of eigenstates.

Also in the process 500, the average phonon energy of the bath in whichthe system characterized by H_(total) is located is calculated (504). Inapproximation, the average phonon energy can be taken as kT, where k isthe Boltzmann constant, and T is the temperature. The average phononenergy can also be calculated in a more precise manner. For example, anopen quantum system model of dynamics, such as the Lindblad formalism,can be selected for the calculation. The selection can be based oncalibration data of the quantum processor. Under the open quantum systemmodel, the average phonon energy of a bath, in which a systemrepresented by H_(total) is located, at any given temperature T can bedefined as:

${\overset{\_}{\omega} = \frac{\Sigma_{0}^{\infty}\omega\;{J(\omega)}{{d\omega}/\left( {e^{\omega/{kT}} - 1} \right)}}{\Sigma_{0}^{\infty}\;{J(\omega)}{{d\omega}/\left( {e^{\omega/{kT}} - 1} \right)}}},$

where J(ω) can be the Omhic spectral density, i.e.,

${{J(\omega)} = {{\lambda\omega}\; e^{- \frac{\omega}{\gamma}}}},$

the super-Omhic spectral density, i.e.,

${{J(\omega)} = {{\lambda\omega}^{3}e^{- \frac{\omega}{\gamma}}}},$

the Drude-Lorentz spectral density, i.e.,

${{J(\omega)} = \frac{2{\lambda\omega\gamma}}{\omega^{2} + \gamma^{2}}},$

or a flat spectral distribution, i.e., J(ω)=1. In these equations, λ isthe reorganization energy and γ is the bath frequency cut-off.

A probability mass function for the ground state fidelity of the QA isselected (506). In some implementations, the probability mass functionis selected manually by a user. Based on the obtained information, thecalculated average phonon energy, and the selected probability massfunction, the process 500 then determines (508) a QG distribution forH_(p). In some implementations, the determination process can be atleast partially performed by a user. For example, the QG distributioncan be represented by an exponential family, such as a Gaussian unitaryensemble, of random matrices selected using a random matrix theorymodel. The average energy level spacing Δg and the maximum and minimumenergy eigenvalues of the QG or

are determined to allow the QG to function as desired. In particular, inthe second phase of the QA schedule, e.g., during time t₁ to t₂ shown inFIG. 3, the average energy level spacing of the QG is chosen such thatthe chosen energy level spacing dominates the energy-level spacing ofthe problem Hamiltonian. The chosen energy level spacing is also muchbigger than the average energy of the phonon bath, e.g., by a factor of5-10, such that the average energy level spacing of the combined QA andQG Δ(g+ε) becomes:

Δ(g+ε)>>ω.

This choice increases the energy level spacing of H_(total) such thatthe combined energy level spacing of H_(tot) is much larger than theaverage phonon energy. Accordingly, possible excitations of the QA to ahigher energy state by thermal fluctuation are suppressed. In addition,the QG is also selected such that in the third phase of the QA schedule,e.g., during time t₂ to t₃ shown in FIG. 3, the average energy levelspacing of the QG leads to:

Δ(g+ε)≈ω  (7)

This choice allows the energy level spacing of H_(total) to be similarto the thermal fluctuation. The QA can relax to a lower energy state orthe ground state at a high rate. The selected exponential family can beparameterized with respect to the controllable parameters, such as thecoupling between qubits, of the quantum hardware.

Alternatively or in addition, a machine learning system can be used totune the control parameters of the QG distribution selected based on therandom matrix theory model. In some implementations, a deep neuralnetwork is used to represent the QG-QA system or the systemcharacterized by H_(tot), and stochastic gradient descent is used totrain the QG distribution. As an example, the training is done byselecting a statistically meaningful number, e.g., 1000, of randommatrices {ε_(im); g_(ijmn)} from a parameterized exponential family thatcan in average generate path-integral Monte-Carlo outputs, within thedesired probability mass function for a given H_(total) of interest. Insome implementations, the training can start with an initial QGdistribution selected based on the desired average combined energy levelspacing Δ(g+ε) discussed above. The initial QG distribution can havepredetermined probability distributions. The training can be supervisedtraining.

The implementation of the random matrix theory model can output agenerative probability mass function. In supervised training, label canbe generated by finding the coupling coefficients of the QG such thatthe probability mass function generated by the QA and the QG has maximumoverlap, e.g., within a given measure or figure of merit such as χ²divergence, with an ideal probability mass function that is known inadvance for the training set. FIG. 6 shows an example process 600 inwhich a control system programs QA hardware, such as a quantumprocessor, for the QA hardware to perform an artificial intelligencetask. The control system includes one or more classical, i.e.,non-quantum, computers, and may also include a quantum computer. Thetask is translated into a machine learning optimization problem, whichis represented in a machine-readable form.

The control system receives (602) the machine-readable machine learningoptimization problem. The control system encodes (606) the optimizationproblem into the energy spectrum of an engineered H_(total). Theencoding is based on structure of the QA hardware, such as the couplingsbetween qubits. An example of H_(total) is the Ising Hamiltonian H_(SG),and the encoding determines the values for the parameters h_(i) andJ_(ij). The encoded information, such as h_(i) and J_(ij), is providedto the QA hardware, which receives (620) the information asinitialization parameters for the hardware. To stabilize the QA during aquantum annealing process to be performed by the QA hardware, thecontrol system further devises (608) a QG, e.g., by selecting one QGfrom a QG distribution determined using the process 500 of FIG. 5. Theselection can be random (pseudo) selection. In some implementations, auser can select the QG from the QG distribution and input the selectionto the control system. The devised QG is characterized by controlparameters including ε_(im) and g_(ijmn), which are sent to the QAhardware to program the QA hardware.

The QA hardware receives (620) the initialization parameters, such ash_(i) and J_(ij), and also receives (622) the control parameters for theQG, such as h_(i) ^(G), J_(ij) ^(G), J_(ij) ^(GA), and is programmed andinitialized by the control system according to the receivedinitialization parameters and QG parameters. The QA hardware implements(624) the quantum annealing schedule to obtain eigenstates of thecombined QA-QG system characterized by H_(tot). The solution to themachine learning optimization problem is encoded in these eigenstates.After a predetermined amount of time, the QA schedule ends and the QAhardware provides (626) an output represented by the eigenstates andtheir corresponding energy spectra. The output can be read by thecontrol system or by another classical computer or quantum computer. Thepredetermined amount of time can be in the order of 1/(Δ(g+ε))².However, shorter or longer periods of time can be used. A shorter timeperiod may provide better quantum speedup, and a longer time period mayprovide a higher ground state fidelity.

As described above, in the output provided by the QA hardware, theground state fidelity of the QA is generally smaller than 1. When thefidelity is smaller than 1, the one-time output provided by the QAhardware may not accurately encode the solution to the problem. In someimplementations, the QA hardware performs the QA schedule multipletimes, using the same QG or different QGs provided by the control systemthat have different sets of control parameters, such as ε_(im) andg_(ijmn), selected from the same QG distribution determined for theproblem, to provide multiple outputs. The multiple outputs can bestatistically analyzed and the problem or the artificial intelligencetask can be resolved or performed based on the statistical results.

In particular, in the process 600, after the control system receives andstores (610) the output provided by the QA hardware, the control systemdetermines (612) whether the QA hardware has completed the predeterminednumber of iterations of QA schedules. If not, then the control systemreturns to the step 608 by devising another QG, which can be the same asthe previously used QG or a different QG selected from the previouslydetermined QG distribution. The QA hardware receives (622) another setof control parameters for the QG and is re-programmed by the controlsystem based on this set of control parameters and the previouslydetermined initialization parameters that encode the problem. The QAschedule is implemented again (624) and another output is provided(626). If the QA hardware has completed the predetermined number ofiterations of QA schedule, then the control system or another dataprocessing system statistically processes (614) all outputs to providesolutions to the problem. Solutions to a problem can be provided with apredetermined degree of certainty that has a sharply peaked PDF about anactual solution to the problem. The PDF can be peaked based on thestatistical analysis.

The predetermined number of iterations can be 100 iterations or more, or1000 iterations or more. In some implementations, the number ofiterations can be chosen in connection with the length of the QAschedule, so that the process 600 can be performed with high efficiencyand provide solutions to the problems with high accuracy. For example,when the length of each QA schedule is relatively short, e.g., shorterthan 1/(Δ(g+ε))², the predetermined number of iterations can be chosento be relatively large, e.g., 1000 iterations or more. In othersituations when the length of each QA schedule is relatively long, e.g.,longer than 1/(Δ(g+ε))², the predetermined number of iterations can bechosen to be relatively small, e.g., less than 1000 iterations.

Embodiments of the digital, i.e., non-quantum, subject matter and thedigital functional operations described in this specification can beimplemented in digital electronic circuitry, in tangibly-embodiedcomputer software or firmware, in computer hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. Embodiments ofthe digital subject matter described in this specification can beimplemented as one or more computer programs, i.e., one or more modulesof computer program instructions encoded on a tangible non-transitorystorage medium for execution by, or to control the operation of, dataprocessing apparatus. The computer storage medium can be amachine-readable storage device, a machine-readable storage substrate, arandom or serial access memory device, or a combination of one or moreof them. Alternatively or in addition, the program instructions can beencoded on an artificially-generated propagated signal, e.g., amachine-generated electrical, optical, or electromagnetic signal, thatis generated to encode information for transmission to suitable receiverapparatus for execution by a data processing apparatus.

The term “data processing apparatus” refers to digital data processinghardware and encompasses all kinds of apparatus, devices, and machinesfor processing data, including by way of example a programmable digitalprocessor, a digital computer, or multiple digital processors orcomputers. The apparatus can also be, or further include, specialpurpose logic circuitry, e.g., an FPGA (field programmable gate array)or an ASIC (application-specific integrated circuit). The apparatus canoptionally include, in addition to hardware, code that creates anexecution environment for computer programs, e.g., code that constitutesprocessor firmware, a protocol stack, a database management system, anoperating system, or a combination of one or more of them.

A computer program, which may also be referred to or described as aprogram, software, a software application, a module, a software module,a script, or code, can be written in any form of programming language,including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a digital computing environment. A computer programmay, but need not, correspond to a file in a file system. A program canbe stored in a portion of a file that holds other programs or data,e.g., one or more scripts stored in a markup language document, in asingle file dedicated to the program in question, or in multiplecoordinated files, e.g., files that store one or more modules,sub-programs, or portions of code. A computer program can be deployed tobe executed on one computer or on multiple computers that are located atone site or distributed across multiple sites and interconnected by adata communication network.

The processes and logic flows described in this specification can beperformed by one or more programmable digital computers, operating withone or more quantum processors, as appropriate, executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA or an ASIC, or by a combination of specialpurpose logic circuitry and one or more programmed computers. For asystem of one or more digital computers to be “configured to” performparticular operations or actions means that the system has installed onit software, firmware, hardware, or a combination of them that inoperation cause the system to perform the operations or actions. For oneor more computer programs to be configured to perform particularoperations or actions means that the one or more programs includeinstructions that, when executed by digital data processing apparatus,cause the apparatus to perform the operations or actions.

Digital computers suitable for the execution of a computer program canbe based on general or special purpose microprocessors or both, or anyother kind of central processing unit. Generally, a central processingunit will receive instructions and data from a read-only memory or arandom access memory or both. The essential elements of a computer are acentral processing unit for performing or executing instructions and oneor more memory devices for storing instructions and data. The centralprocessing unit and the memory can be supplemented by, or incorporatedin, special purpose logic circuitry. Generally, a digital computer willalso include, or be operatively coupled to receive data from or transferdata to, or both, one or more mass storage devices for storing data,e.g., magnetic, magneto-optical disks, or optical disks. However, acomputer need not have such devices.

Computer-readable media suitable for storing computer programinstructions and data include all forms of non-volatile memory, mediaand memory devices, including by way of example semiconductor memorydevices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks,e.g., internal hard disks or removable disks; magneto-optical disks; andCD-ROM and DVD-ROM disks.

Control of the various systems described in this specification, orportions of them, can be implemented in a computer program product thatincludes instructions that are stored on one or more non-transitorymachine-readable storage media, and that are executable on one or moredigital processing devices. The systems described in this specification,or portions of them, can each be implemented as an apparatus, method, orelectronic system that may include one or more digital processingdevices and memory to store executable instructions to perform theoperations described in this specification.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular embodiments. Certain features that are described in thisspecification in the context of separate embodiments can also beimplemented in combination in a single embodiment. Conversely, variousfeatures that are described in the context of a single embodiment canalso be implemented in multiple embodiments separately or in anysuitable subcombination. Moreover, although features may be describedabove as acting in certain combinations and even initially claimed assuch, one or more features from a claimed combination can in some casesbe excised from the combination, and the claimed combination may bedirected to a subcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the embodiments described above should not beunderstood as requiring such separation in all embodiments, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular embodiments of the subject matter have been described. Otherembodiments are within the scope of the following claims. For example,the actions recited in the claims can be performed in a different orderand still achieve desirable results. As one example, the processesdepicted in the accompanying figures do not necessarily require theparticular order shown, or sequential order, to achieve desirableresults. In some cases, multitasking and parallel processing may beadvantageous.

1. A method comprising: for a quantum processor solving an optimizationproblem, encoding, by one or more processors, the optimization probleminto an energy spectrum of a quantum Hamiltonian H_(total) thatcharacterizes the quantum processor, wherein the quantum processor iscontrollable such that H_(total) evolves from an initial quantumHamiltonian H_(i) over time to a problem quantum Hamiltonian H_(p),wherein an energy spectrum of H_(p) encodes a solution to theoptimization problem, and wherein a quantum state evolves from a groundstate of H_(i) towards a ground state of H_(p) as H_(total) evolves fromH_(i) to H_(p); outputting a first set of control parameters based onthe encoding for programming the quantum processor and parameterizationof H_(total); outputting a second set of control parameters forprogramming the quantum processor to reflect the parameterization ofH_(total) to limit evolution of the quantum state into the ground stateof H_(p), as H_(total) evolves to H_(p) without diagonalizing H_(p); andreceiving, by the one or more processors from the quantum processor, anoutput associated with a state of the quantum state at the end of theevolution.